2004 Chevy Silverado Continues to blow hot air

Is this an auto temp control system?
Was it good and cold before changing the battery?

That really doesn't tell me if you have an auto system or not. If you set it to a specific temp and it automatically controls the blower speed and temp, then it's an auto system.

The auto system in these has to go through a calibration of the blend door actuators whenever the battery is disconnected. You disconnect the battery to start over and leave the key on for a minute or two. If it does not immediately swing to full cold and full heat then settle, then the blend door may have gotten stuck at one end. This can be a real big problem because the entire dash will have to be removed to access it.

That's a manual system but it's likely still an air door problem. You can try pinching off one of the heater hoses to verify this.

They make special pliers for doing that. A pair of locking needle nose will work too.